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{{Short description|Kind of spacecraft trajectory}}
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[[File:Aldrin Cycler.gif|thumb|A Mars cycler is an elliptical orbit (green) that crosses the orbits of Earth (blue) and Mars (red), and encounters both planets at the points where it crosses their orbits, although not necessarily on every orbit. (Not to scale.)|upright=1.3]]
A '''Mars cycler''' (or '''Earth–Mars cycler''') is a kind of [[cycler]], a spacecraft with a [[trajectory]] that encounters [[Earth]] and [[Mars]] regularly. The '''Aldrin cycler''' is an example of a Mars cycler.

Cyclers are potentially useful for transporting people or materials between those bodies using minimal propellant (relying on [[Gravity assist|gravity-assist]] flybys for most trajectory changes) and can carry heavy [[Radiation protection|radiation shielding]] to protect people in transit from [[cosmic ray]]s and [[Geomagnetic storm|solar storms]].

==Earth–Mars cyclers==
A cycler is a [[Interplanetary spaceflight#Cyclers|trajectory]] that encounters two or more bodies regularly. Once the [[orbit]] is established, no propulsion is required to shuttle between the two, although some minor corrections may be necessary due to small perturbations in the orbit. The use of cyclers was considered in 1969 by Walter M. Hollister, who examined the case of an Earth–Venus cycler.{{sfn|Hollister|1969|p=366}}  Hollister did not have any particular mission in mind, but posited their use for both regular communication between two planets, and for multi-planet flyby missions.{{sfn|Hollister|1969|p=369}}

A Martian year is 1.8808 Earth years, so Mars makes eight orbits of the [[Sun]] in about the same time as Earth makes 15. Cycler trajectories between Earth and Mars occur in whole-number multiples of the [[synodic period]] between the two planets, which is about 2.135 Earth years.{{sfn|Byrnes|Longuski|Aldrin|1993|p=334}} In 1985, [[Buzz Aldrin]] presented an extension of his earlier [[Lunar cycler]] work which identified a Mars cycler corresponding to a single synodic period.{{sfn|Aldrin|1985|pp=3–10}} The Aldrin cycler (as it is now known) makes a single eccentric loop around the Sun. It travels from Earth to Mars in 146 days (4.8 months), spends the next 16 months beyond the orbit of Mars, and takes another 146 days going from the orbit of Mars back to the first crossing of Earth's orbit.{{sfn|McConaghy|Longuski|Byrnes|2002|p=6}}

The existence of the now-eponymous Aldrin cycler was calculated and confirmed by scientists at [[Jet Propulsion Laboratory]] later that year, along with the VISIT-1 and VISIT-2 cyclers proposed by John Niehoff in 1985.{{sfn|Friedlander|Niehoff|Byrnes|Longuski|1986|p=31}}{{sfn|McConaghy|Longuski|Byrnes|2002|p=1}} For each Earth–Mars cycler that is not a multiple of seven synodic periods, an outbound cycler intersects Mars on the way out from Earth while an inbound cycler intersects Mars on the way in to Earth. The only difference in these trajectories is the date in the synodic period in which the vehicle is launched from Earth. Earth–Mars cyclers with a multiple of seven synodic periods return to Earth at nearly the same point in its orbit and may encounter Earth and/or Mars multiple times during each cycle. VISIT-1 encounters Earth three times and Mars four times in 15 years. VISIT-2 encounters Earth five times and Mars two times in 15 years.{{sfn|McConaghy|Longuski|Byrnes|2002|p=6}} Some possible Earth–Mars cyclers include the following:{{sfn|McConaghy|Longuski|Byrnes|2002|p=6}}
{| class="wikitable sortable"
|-
! scope=col| Synodic periods per cycle
! scope=col| Solar revolutions per cycle
! scope=col| Time per cycle (years)
! scope=col| [[Aphelion]] radius (AU)
! scope=col| Earth/Mars transfer time (days)
! scope=col| Notes
|-
| 1
| 1
| 2.135
| 2.23
| 146
| Aldrin cycler
|-
| 2
| 2
| 4.27
| 2.33
| 158
|
|-
| 2
| 3
| 4.27
| 1.51
| 280
| Aphelion inside semi-major axis of Mars orbit
|-
| 3
| 4
| 6.405
| 1.89
| 189
|
|-
| 3
| 5
| 6.405
| 1.45
| 274
| Aphelion inside semi-major axis of Mars orbit
|-
| 3
| 5
| 6.405
| 1.52
| 134
| Aphelion inside semi-major axis of Mars orbit
|-
| 4
| 5
| 8.54
| 1.82
| 88
|
|-
| 4
| 6
| 8.54
| 1.53
| 157
| Aphelion inside aphelion of Mars orbit
|-
| 5
| 4
| 10.675
| 2.49
| 75
|
|-
| 5
| 5
| 10.675
| 2.09
| 89
|
|-
| 5
| 6
| 10.675
| 1.79
| 111
|
|-
| 5
| 7
| 10.675
| 1.54
| 170
| Aphelion inside aphelion of Mars orbit
|-
| 5
| 8
| 10.675
| 1.34
| 167
| Aphelion inside semi-major axis of Mars orbit
|-
| 6
| 4
| 12.81
| 2.81
| 87
|
|-
| 6
| 5
| 12.81
| 2.37
| 97
|
|-
| 6
| 6
| 12.81
| 2.04
| 111
|
|-
| 6
| 7
| 12.81
| 1.78
| 133
| Requires minimal ballistic correction
|-
| 6
| 8
| 12.81
| 1.57
| 179
| Requires minimal ballistic correction
|-
| 6
| 9
| 12.81
| 1.40
| 203
| Aphelion inside semi-major axis of Mars orbit; Requires minimal ballistic correction
|}

A detailed survey of Earth–Mars cycler trajectories was conducted by Ryan Russell and Cesar Ocampo from the [[University of Texas at Austin]], [[Texas]]. They identified 24 Earth-Mars cyclers with periods of two to four synodic periods, and 92 cyclers with periods of five or six synodic periods. They also found hundreds of non-ballistic cyclers, ones which would require some powered maneuvers.{{sfn|Russell|Ocampo|2004|p=321}}

==Physics==
[[File:Gravity assist velocity diagram.png|thumb|Gravity assist velocity diagram.]]

Earth orbits the Sun in one Earth year, Mars in 1.881. Neither orbit is perfectly circular; Earth has an [[orbital eccentricity]] of 0.0168, and Mars of 0.0934. The two orbits are not quite coplanar either, as the orbit of Mars is inclined by 1.85 [[degree (angle)|degrees]] to that of Earth. The effect of the gravity of Mars on the cycler orbits is almost negligible, but that of the far more massive Earth needs to be considered. If we ignore these factors, and approximate Mars's orbital period as 1.875 Earth years, then 15 Earth years is 8 Martian years. In the diagram above, a spacecraft in an Aldrin cycler orbit that starts from Earth at point E1 will encounter Mars at M1. When it gets back to E1 just over two Earth years later, Earth will no longer be there, but it will encounter Earth again at E2, which is <math>51.4^\circ</math>, {{frac|1|7}} of an Earth orbit, further round.{{sfn|Byrnes|Longuski|Aldrin|1993|pp=334–335}}

The shape of the cycler orbit can be obtained from the [[conic equation]]:

<math display="block">r = a\frac{1-\epsilon^2}{1+\epsilon\cos\theta}</math>

where <math>r</math> is 1 [[astronomical unit]], <math>a</math> is the [[semi-major and semi-minor axes|semi-major axis]], <math>\epsilon</math> is the orbital eccentricity and <math>\theta=-25.7^\circ</math> (half of <math>-51.4^\circ</math>). We can obtain <math>a</math> by solving [[Lambert's problem]] with <math>51.4^\circ</math> as the initial and final transfer angle. This gives:

<math display="block">a = 1.60</math>

Solving the [[quadratic equation]] gives:

<math display="block">\epsilon = 0.393</math>

with an orbital period of 2.02 years.{{sfn|Byrnes|Longuski|Aldrin|1993|pp=334–335}}

The angle at which the spacecraft flies past Earth, <math>\gamma</math>, is given by:

<math display="block">\tan \gamma = \frac{\epsilon r\sin\theta}{a(1-\epsilon^2)}</math>

Substituting the values given and derived above gives a value for <math>\gamma</math> of <math>7.18^\circ</math>. We can calculate the [[gravity assist]] from Earth:

<math display="block">\Delta V = 2 V \sin \gamma</math>

where <math>V</math> is the heliocentric flyby velocity. This can be calculated from:

<math display="block">V = V_E\sqrt{2-\frac{r}{a}}</math>

where {{SubSup|V|E}} is the velocity of Earth, which is 29.8&nbsp;km/s. Substituting gives us V = 34.9&nbsp;km/s, and {{DELTA}}V = 8.73&nbsp;km/s.{{sfn|Byrnes|Longuski|Aldrin|1993|pp=334–335}}

The excess speed is given by:

<math display="block">V_\infty = \sqrt{V^2 + V_E^2 -2 V V_E \cos \gamma}</math>

Which gives a value for {{SubSup|V|∞}} of 6.54&nbsp;km/s. The turn angle <math>\delta</math> can be calculated from:

<math display="block">\Delta V = 2 V_\infty \sin \delta</math>

Which gives <math>\delta = 41.9^\circ</math>, meaning that we have an <math>83.8^\circ</math> turn. The radius of closest approach to Earth {{SubSup|r|p}} will be given by:

<math display="block">\sin \delta = \frac{1}{1+\frac{r_pV_\infty^2}{\mu_E}}</math>

Where {{SubSup|μ|E}} is the [[gravitational constant]] of the Earth. Substituting the values gives  {{SubSup|r|p}} = {{convert|4640|km}}, which is bad because the radius of the Earth is {{convert|6371|km}}. A correction would therefore be required to comfortably avoid the planet.{{sfn|Byrnes|Longuski|Aldrin|1993|pp=334–335}}

==Proposed use==
[[File:Aldrin Cycler 2.gif|thumb|Rather than wait for the cycler to come around again, Aldrin proposed using a second cycler to make the return trip. (Not to scale.)]]
Aldrin proposed a pair of Mars cycler vehicles providing regular transport between Earth and Mars.{{sfn |Aldrin |1985 |pp=3–10}} While astronauts can tolerate traveling to the Moon in relatively cramped spacecraft for a few days, a mission to Mars, lasting several months, would require much more habitable accommodations for the much longer journey: astronauts would need a facility with ample living space, life support, and heavy radiation shielding.{{sfn |Friedlander |Niehoff |Byrnes |Longuski |1986 |p=31}}<ref name="Roadmap">{{cite magazine |last1=Aldrin |first1=Buzz |author-link1=Buzz Aldrin |first2=David |last2=Noland |date=13 December 2005 |title=Buzz Aldrin's roadmap to Mars – A ''PM'' exclusive |magazine=[[Popular Mechanics]] |url=http://www.popularmechanics.com/space/moon-mars/a333/2076326/ }}</ref> A 1999 NASA study estimated that a mission to Mars would require lifting about {{convert|437|t|ST|sp=us}} into space, of which {{convert|250|t|ST|sp=us}} was propellant.<ref name="Damn Interesting"/>

Aldrin proposed that the costs of Mars missions could be greatly reduced by use of large [[space station]]s in cyclic orbits called ''castles''. Once established in their orbits, they would make regular trips between Earth and Mars without requiring any propellant. Other than consumables, cargo would therefore only have to be launched once.{{sfn |Friedlander |Niehoff |Byrnes |Longuski |1986 |p=31}}<ref name="Roadmap"/> Two ''castles'' would be used, an outbound one on an Aldrin ''cycler'' with a fast transfer to Mars and long trip back, and an inbound one with fast trip to Earth and long return to Mars,{{sfn|Byrnes|Longuski|Aldrin|1993|p=334}} which Aldrin called ''up and down escalators''.{{sfn|Friedlander|Niehoff|Byrnes|Longuski|1986|p=31}}

The astronauts would meet up with the cycler in Earth orbit and later Mars orbit in specialised craft called ''taxis''. One cycler would travel an outbound route from Earth to Mars in about five months. Another Mars cycler in a complementary trajectory would travel from Mars to Earth, also in about five months. Taxi and cargo vehicles{{efn|Most cargo would be sent directly to Mars, as the advantages of a cycler (living space, shielding, and life support systems) are of value primarily to human passengers.}} would attach to the cycler at one planet and detach upon reaching the other.<ref name="Damn Interesting">{{cite web |last=Bellows |first=Alan |date=10 April 2008 |title=The Martian Express |website=Damn Interesting |url=http://www.damninteresting.com/?p=960 |archive-url=https://web.archive.org/web/20220901111922/https://www.damninteresting.com/the-martian-express/ |archive-date=1 September 2022 |url-status=live}}</ref> The cycler concept would therefore provide for routine, safe, and economical transport between Earth and Mars.{{sfn |Rogers |Hughes |Longuski |Aldrin |2015 |p=114}}

A significant drawback of the ''cycler'' concept was that the Aldrin cycler flies by both planets at high speed. A taxi would need to accelerate to {{convert|15000|mph|km/s}} around Earth, and {{convert|22000|mph|km/s}} near Mars. To get around this, Aldrin proposed what he called a ''semi-cycler'', in which the ''castle'' would slow down around Mars, orbiting it, and later resume the ''cycler'' orbit. This would require fuel to execute the braking and re-cycling maneuvers.<ref name="Roadmap"/><ref name="Damn Interesting"/>

The castles could be inserted into cycler orbits with considerable savings in fuel by performing a series of low thrust maneuvers:{{sfn |Rogers |Hughes |Longuski |Aldrin |2015 |p=114}} The castle would be placed into an interim orbit upon launch, and then use an Earth-[[Gravity assist|swing-by maneuver]] to boost it into the final cycler orbit.{{sfn |Rogers |Hughes |Longuski |Aldrin |2015 |pp=120–121}} Assuming the use of conventional fuels,{{efn|Conventional propellants in the vicinity of Earth are [[liquid hydrogen]] and [[liquid oxygen]], which can be refrigerated on the ground and used during or soon after launch. The 2×[[liquid hydrogen|{{chem|H|2}}]] + [[liquid oxygen|{{chem|O|2}}]] combination has a [[specific impulse]] of about {{convert|450|isp|km/s|abbr=on}}. Deep space maneuvers use toxic, non-cryogenic [[monomethylhydrazine|mono-methyl hydrazine]] and [[nitrogen tetroxide]] propellants (e.g. used by the [[Galileo (spacecraft)|''Galileo'' spacecraft]]) with a {{convert|300|isp|km/s|abbr=on}} specific impulse. Safer and more efficient [[cryogenic fuel]]s like [[liquid hydrogen|{{chem|O|2}}]] and [[liquid hydrogen|{{chem|H|2}}]] cannot be economically carried into deep space: without prohibitively massive refrigeration they boil away.}} it is possible to estimate the fuel required to establish a cycler orbit.{{sfn |Rogers |Hughes |Longuski |Aldrin |2015|p=123}} In the case of the Aldrin cycler, use of a gravity assist reduces the fuel requirement by about {{convert|24.3|t|ST|sp=us}}, or 15&nbsp;percent. Other cyclers showed less impressive improvement, due to the shape of their orbits, and when they encounter the Earth. In the case of the VISIT-1 cycler, the benefit would be around {{convert|0.2|t|ST|sp=us}}, less than one percent, which would hardly justify the additional three years required to establish the orbit.{{sfn |Rogers |Hughes |Longuski |Aldrin |2015 |p=123}}

==See also==
*[[Permanence (novel)#Interstellar cycler|Interstellar cycler]]
*[[Lunar cycler]]

==Notes==
{{notelist}}

==References==
{{reflist}}

===Additional references===
* {{cite web |first=Buzz |last=Aldrin |author-link1=Buzz Aldrin |title=Cyclic Trajectory Concepts   |date=28 October 1985 |url=http://buzzaldrin.com/files/pdf/1985.10.28.ALDRIN_SAIC_PAPER.Cyclic_Trajectory_Concepts_0.pdf|website=buzzaldrin.com |access-date=4 August 2019 |archive-url=https://web.archive.org/web/20180731213226/http://buzzaldrin.com/files/pdf/1985.10.28.ALDRIN_SAIC_PAPER.Cyclic_Trajectory_Concepts_0.pdf |archive-date=31 July 2018 |url-status=dead }}
* {{cite journal |last1=Byrnes |first1=Dennis V. |last2=Longuski |first2=James M. |last3=Aldrin |first3=Buzz |author-link3=Buzz Aldrin |title=Cycler orbit between Earth and Mars |journal=Journal of Spacecraft and Rockets |volume=30 |issue=3 |year=1993 |pages=334–336 |doi=10.2514/3.25519 |bibcode=1993JSpRo..30..334B }}
*{{cite conference |last1=Friedlander |first1= Alan L. |last2=Niehoff |first2=John C. |last3=Byrnes |first3=Dennis V. |last4=Longuski |first4=James M. |title=Circulating Transportation Orbits between Earth and Mars |id=86-2009|conference=Astrodynamics Conference|location=Williamsburg, Virginia|date=August 18–20, 1986|publisher=[[American Institute of Aeronautics and Astronautics]]|url=https://engineering.purdue.edu/people/james.m.longuski.1/ConferencePapersPresentations/1986CirculatingTransportationOrbitsbetweenEarthandMars.pdf |access-date=4 August 2019 |doi=10.2514/6.1986-2009 |archive-url=https://web.archive.org/web/20140706005927/https://engineering.purdue.edu/people/james.m.longuski.1/ConferencePapersPresentations/1986CirculatingTransportationOrbitsbetweenEarthandMars.pdf |url-status=dead |archive-date=6 July 2014}}
* {{cite journal |last=Hollister |first=W. M. |title=Periodic orbits for interplanetary flight |journal=Journal of Spacecraft and Rockets |issn= 0022-4650 |volume=6 |issue=4 |year=1969 |pages=366–369 |doi=10.2514/3.29664 |bibcode=1969JSpRo...6..366H }} }
* {{cite journal |last1=McConaghy  |first1=T. Troy |last2=Longuski |first2=James M. |last3=Byrnes  |first3=Dennis V. |title=Analysis of a Broad Class of Earth-Mars Cycler Trajectories |journal=American Institute of Aeronautics and Astronautics |id=2002–4420 |year=2002 |url=https://engineering.purdue.edu/people/james.m.longuski.1/ConferencePapersPresentations/2002AnalysisofaBroadClassofEarth-MarsCyclerTrajectories.pdf |archive-url=https://web.archive.org/web/20220206061137/https://engineering.purdue.edu/people/james.m.longuski.1/ConferencePapersPresentations/2002AnalysisofaBroadClassofEarth-MarsCyclerTrajectories.pdf |url-status=dead |archive-date=6 February 2022}}
* {{cite journal |last1=Rogers |first1=Blake A. |last2=Hughes |first2=Kyle M. |last3=Longuski |first3=James M. |last4= Aldrin |first4= Buzz |author-link4=Buzz Aldrin |title=Establishing Cycler Trajectories between Earth and Mars |journal=Acta Astronautica |issn=0094-5765 |year=2015 |volume=112 |pages=114–125 |doi=10.1016/j.actaastro.2015.03.002 |bibcode=2015AcAau.112..114R }}
* {{cite journal |last1=Russell |first1=Ryan |last2=Ocampo |first2=Cesar  |title=Systematic Method for Constructing Earth-Mars Cyclers Using Free-Return Trajectories |journal=Journal of Guidance, Control, and Dynamics |volume=27 |issue=3 |year=2004 |pages=321–335 |doi=10.2514/1.1011 |bibcode=2004JGCD...27..321R }}

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